Mathematics 10 — Exercise 5.4 · Q1 (i)

Full question

Solve the following equation by reducing it to a quadratic equation. Check the resulting values in the original equation.

Step-by-step solution

1. Let t = x², so x⁴ = t².

   $$t=x^2$$

2. After substitution the quadratic equation is:

   $$5\times t^2-19\times t+12=0$$

3. Substitute the function expressions, keeping the brackets.

   $$5\times t^2-19\times t+12-0$$

4. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

   $$5\times t^2-19\times t+12$$

5. Write EVERY distributed product before collecting terms.

   $$12-19\times t+5\times t^2-0$$

6. Calculate this individual product, including its sign and powers.

   $$12=12$$

7. Calculate this individual product, including its sign and powers.

   $$-19\times t=-19\times t$$

8. Calculate this individual product, including its sign and powers.

   $$5\times t^2=5\times t^2$$

9. Calculate this individual product, including its sign and powers.

   $$\left(-1\right)\times 0=0$$

10. These terms have the same variable part. Add ONLY their coefficients.

    $$12+0=12$$

11. These terms have the same variable part. Add ONLY their coefficients.

    $$-19=-19$$

12. These terms have the same variable part. Add ONLY their coefficients.

    $$5=5$$

13. Put the simplified groups together.

    $$5\times t^2-19\times t+12$$

14. Collect terms with the same power and simplify their coefficients.

    $$5\times t^2-19\times t+12$$

15. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

    $$5\times t^2-19\times t+12=0$$

16. Split the middle coefficient into two numbers whose sum is b and whose product is a·c.

    $$(-4-15=-19,\left(-4\right)\times \left(-15\right)=60)$$

17. Rewrite the middle term using those two parts.

    $$5\times t^2-4\times t-15\times t+12=0$$

18. Group the first two terms and the last two. Take out the common factor from each group.

    $$t\times \left(5\times t-4\right)-3\times \left(5\times t-4\right)=0$$

19. The common bracket can now be factored out.

    $$\left(t-3\right)\times \left(5\times t-4\right)=0$$

20. A product is zero only if at least one factor is zero. Set each linear factor equal to zero.

    $$(t-3=0,5\times t-4=0)$$

21. Take this factor equal to zero.

    $$t-3=0$$

22. Subtract the constant from both sides.

    $$t-3+3=0+3$$

23. Simplify both sides.

    $$t=3$$

24. Divide both sides by the coefficient of the variable.

    $$t=3\times \frac{1}{1}$$

25. Reduce the fraction to obtain this root.

    $$t=3$$

26. Take this factor equal to zero.

    $$5\times t-4=0$$

27. Subtract the constant from both sides.

    $$5\times t-4+4=0+4$$

28. Simplify both sides.

    $$5\times t=4$$

29. Divide both sides by the coefficient of the variable.

    $$t=\frac{4}{5}$$

30. Reduce the fraction to obtain this root.

    $$t=\frac{4}{5}$$

31. Substitute this root into the original equation before simplifying.

    $$0=0$$

32. Calculate both sides; their equal values confirm the root.

    $$0=0$$

33. Substitute this root into the original equation before simplifying.

    $$0=0$$

34. Calculate both sides; their equal values confirm the root.

    $$0=0$$

35. Substitute this t value back into x²=t.

    $$x^2=\frac{4}{5}$$

36. A denominator cannot equal zero. Keep the original excluded values throughout the calculation.

    $$5\ne 0$$

37. Multiply BOTH sides by the common denominator to remove the fractions.

    $$5\times x^2=5\times \frac{4}{5}$$

38. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

    $$5\times x^2-4=0$$

39. Integer factorization does not apply here, so use the quadratic formula.

40. Calculate this operation before combining the remaining terms.

    $$20\times \left(-4\right)=-80$$

41. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \left(-80\right)=80$$

42. Compute the discriminant one operation at a time: square b, multiply 4,a,c, then subtract.

    $$0^2-20\times \left(-4\right)$$

43. Calculate this operation before combining the remaining terms.

    $$0^2=0$$

44. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times 5\times \left(-4\right)=80$$

45. Calculate this operation before combining the remaining terms.

    $$0+80=80$$

46. Compare with ax² + bx + c = 0 and read a, b, c WITH their signs.

    $$(a=5,b=0,c=-4)$$

47. Compute D = b² − 4ac. A minus times a minus becomes a plus.

    $$D=0^2-20\times \left(-4\right)$$

48. Simplify the discriminant.

    $$D=80$$

49. Use x = (−b ± √D)/(2a). Show the plus and minus cases separately.

    $$(x=\frac{0+4\times \sqrt{5}}{10},x=\frac{0-4\times \sqrt{5}}{10})$$

50. Simplify each numerator and divide by 2a.

    $$\{\left(-\frac{2}{5}\right)\times \sqrt{5},\frac{2}{5}\times \sqrt{5}\}$$

51. Calculate this root branch separately, preserving the sign.

    $$x=\frac{0+1\times \sqrt{80}}{2\times 5}$$

52. Extract a perfect-square factor from under the root.

    $$80=16\times 5$$

53. Use √(ab)=√a×√b for nonnegative a,b; take the root of the perfect square.

    $$\sqrt{80}=4\times \sqrt{5}$$

54. Calculate this operation before combining the remaining terms.

    $$\sqrt{80}=4\times \sqrt{5}$$

55. Calculate this operation before combining the remaining terms.

    $$1\times 4\times \sqrt{5}=4\times \sqrt{5}$$

56. Calculate this operation before combining the remaining terms.

    $$0+4\times \sqrt{5}=4\times \sqrt{5}$$

57. Calculate this operation before combining the remaining terms.

    $$2\times 5=10$$

58. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 10}$$

59. Calculate these two products separately.

    $$\frac{1}{10}$$

60. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{10}=\frac{1}{10}$$

61. Calculate this operation before combining the remaining terms.

    $$4\times \sqrt{5}\times \frac{1}{10}=\frac{2}{5}\times \sqrt{5}$$

62. Calculate this root branch separately, preserving the sign.

    $$x=\frac{0-\sqrt{80}}{2\times 5}$$

63. Extract a perfect-square factor from under the root.

    $$80=16\times 5$$

64. Use √(ab)=√a×√b for nonnegative a,b; take the root of the perfect square.

    $$\sqrt{80}=4\times \sqrt{5}$$

65. Calculate this operation before combining the remaining terms.

    $$\sqrt{80}=4\times \sqrt{5}$$

66. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 4\times \sqrt{5}=\left(-4\right)\times \sqrt{5}$$

67. Calculate this operation before combining the remaining terms.

    $$0-4\times \sqrt{5}=\left(-4\right)\times \sqrt{5}$$

68. Calculate this operation before combining the remaining terms.

    $$2\times 5=10$$

69. Calculate this operation before combining the remaining terms.

    $$\left(\left(-4\right)\times \sqrt{5}\right)\times \frac{1}{10}=\left(-\frac{2}{5}\right)\times \sqrt{5}$$

70. Substitute this root into the original equation before simplifying.

    $$\frac{4}{5}=\frac{4}{5}$$

71. Calculate both sides; their equal values confirm the root.

    $$\frac{4}{5}=\frac{4}{5}$$

72. Substitute this root into the original equation before simplifying.

    $$\frac{4}{5}=\frac{4}{5}$$

73. Calculate both sides; their equal values confirm the root.

    $$\frac{4}{5}=\frac{4}{5}$$

74. Substitute this t value back into x²=t.

    $$x^2=3$$

75. Substitute the function expressions, keeping the brackets.

    $$x^2-3$$

76. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

    $$x^2-3$$

77. Write EVERY distributed product before collecting terms.

    $$x^2-3$$

78. Calculate this individual product, including its sign and powers.

    $$x^2=x^2$$

79. Calculate this individual product, including its sign and powers.

    $$\left(-1\right)\times 3=-3$$

80. These terms have the same variable part. Add ONLY their coefficients.

    $$1=1$$

81. These terms have the same variable part. Add ONLY their coefficients.

    $$-3=-3$$

82. Put the simplified groups together.

    $$x^2-3$$

83. Collect terms with the same power and simplify their coefficients.

    $$x^2-3$$

84. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

    $$x^2-3=0$$

85. Integer factorization does not apply here, so use the quadratic formula.

86. Calculate this operation before combining the remaining terms.

    $$4\times \left(-3\right)=-12$$

87. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times \left(-12\right)=12$$

88. Compute the discriminant one operation at a time: square b, multiply 4,a,c, then subtract.

    $$0^2-4\times \left(-3\right)$$

89. Calculate this operation before combining the remaining terms.

    $$0^2=0$$

90. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times 1\times \left(-3\right)=12$$

91. Calculate this operation before combining the remaining terms.

    $$0+12=12$$

92. Compare with ax² + bx + c = 0 and read a, b, c WITH their signs.

    $$(a=1,b=0,c=-3)$$

93. Compute D = b² − 4ac. A minus times a minus becomes a plus.

    $$D=0^2-4\times \left(-3\right)$$

94. Simplify the discriminant.

    $$D=12$$

95. Use x = (−b ± √D)/(2a). Show the plus and minus cases separately.

    $$(x=\frac{0+2\times \sqrt{3}}{2},x=\frac{0-2\times \sqrt{3}}{2})$$

96. Simplify each numerator and divide by 2a.

    $$\{\left(-1\right)\times \sqrt{3},\sqrt{3}\}$$

97. Calculate this root branch separately, preserving the sign.

    $$x=\frac{0+1\times \sqrt{12}}{2\times 1}$$

98. Extract a perfect-square factor from under the root.

    $$12=4\times 3$$

99. Use √(ab)=√a×√b for nonnegative a,b; take the root of the perfect square.

    $$\sqrt{12}=2\times \sqrt{3}$$

100. Calculate this operation before combining the remaining terms.

     $$\sqrt{12}=2\times \sqrt{3}$$

101. Calculate this operation before combining the remaining terms.

     $$1\times 2\times \sqrt{3}=2\times \sqrt{3}$$

102. Calculate this operation before combining the remaining terms.

     $$0+2\times \sqrt{3}=2\times \sqrt{3}$$

103. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

104. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times 1}{1\times 2}$$

105. Calculate these two products separately.

     $$\frac{1}{2}$$

106. Calculate this operation before combining the remaining terms.

     $$1\times \frac{1}{2}=\frac{1}{2}$$

107. Calculate this operation before combining the remaining terms.

     $$2\times \sqrt{3}\times \frac{1}{2}=\sqrt{3}$$

108. Calculate this root branch separately, preserving the sign.

     $$x=\frac{0-\sqrt{12}}{2\times 1}$$

109. Extract a perfect-square factor from under the root.

     $$12=4\times 3$$

110. Use √(ab)=√a×√b for nonnegative a,b; take the root of the perfect square.

     $$\sqrt{12}=2\times \sqrt{3}$$

111. Calculate this operation before combining the remaining terms.

     $$\sqrt{12}=2\times \sqrt{3}$$

112. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 2\times \sqrt{3}=\left(-2\right)\times \sqrt{3}$$

113. Calculate this operation before combining the remaining terms.

     $$0-2\times \sqrt{3}=\left(-2\right)\times \sqrt{3}$$

114. Calculate this operation before combining the remaining terms.

     $$2\times 1=2$$

115. Calculate this operation before combining the remaining terms.

     $$\left(\left(-2\right)\times \sqrt{3}\right)\times \frac{1}{2}=\left(-1\right)\times \sqrt{3}$$

116. Substitute this root into the original equation before simplifying.

     $$3=3$$

117. Calculate both sides; their equal values confirm the root.

     $$3=3$$

118. Substitute this root into the original equation before simplifying.

     $$3=3$$

119. Calculate both sides; their equal values confirm the root.

     $$3=3$$

120. Collect every valid root from every substitution branch.

     $$\{\left(-1\right)\times \sqrt{3},\sqrt{3},\left(-\frac{2}{5}\right)\times \sqrt{5},\frac{2}{5}\times \sqrt{5}\}$$

Answer

The final answer is shown below.